Lattice Boltzmann model for resistive relativistic magnetohydrodynamics

In this paper, we develop a lattice Boltzmann model for relativistic magnetohydrodynamics (MHD). Even though the model is derived for resistive MHD, it is shown that it is numerically robust even in the high conductivity (ideal MHD) limit. In order to validate the numerical method, test simulations are carried out for both ideal and resistive limits, namely the propagation of Alfvén waves in the ideal MHD and the evolution of current sheets in the resistive regime, where very good agreement is observed comparing to the analytical results. Additionally, two-dimensional magnetic reconnection driven by Kelvin-Helmholtz instability is studied and the effects of different parameters on the reconnection rate are investigated. It is shown that the density ratio has a negligible effect on the magnetic reconnection rate, while an increase in shear velocity decreases the reconnection rate. Additionally, it is found that the reconnection rate is proportional to ${\sigma }^{-1/2}, \sigma$ being the conductivity, which is in agreement with the scaling law of the Sweet-Parker model. Finally, the numerical model is used to study the magnetic reconnection in a stellar flare. Three-dimensional simulation suggests that the reconnection between the background and flux rope magnetic lines in a stellar flare can take place as a result of a shear velocity in the photosphere.

Figure 1

The D3Q19 lattice configuration for the LB model to solve the relativistic fluid equations.

Figure 2

The configuration of the auxiliary vectors for the LB model to solve the Maxwell equations.

Figure 3

Results of the numerical test for the propagation of Alfvén waves in the limit of ideal MHD ($\sigma ={10}^5$) for (a) magnetic field in $z$ direction and (b) electric field in $x$ direction. For both plots, the numerical results are shown at $t=1$ (blue circle symbols) and at $t=1.5$ (red square symbols). The dashed black line shows the initial condition and solid black lines are the exact analytical solutions at the corresponding time.

Figure 4

Results of the numerical test for the evolution of a current sheet in the resistive regime. Dashed green line (upper curve at $x>0$) and dashed red line (second upper curve at $x>0$) show the initial condition of the current sheet for $\sigma =100$ and $\sigma =50$, respectively, which corresponds to the analytical solution, Eq. (57), at $t=1$. The green triangle and red square symbols show the result of the numerical solution at $t=9$, for $\sigma =100$ and $\sigma =50$, respectively. The solid black lines show the exact analytical solution at this time.

Figure 5

Results of the convergence test for the simulation of the evolution of a current sheet in the resistive regime with $\sigma =100$ in a log-log plot. Symbols show the numerical error and the solid blue line is the fitted line with slope $-1.9668$, which shows a second order convergence. In the inset the speedup obtained by using the OpenMP parallelizing method is reported vs the number of threads. The dashed line shows the ideal speedup.

Figure 6

Snapshots of the density for the KH instability at times (a) $t=0.0$ (initial condition), (b) $t=3.31$, (c) $t=6.62$, (d) $t=15.47$, for the case with $\sigma =100, \mathrm{\Delta }n=1.8, U_0=0.6c$. The white lines show the magnetic field lines.

Figure 7

Reconnection rate vs time for the case $\sigma =100, \mathrm{\Delta }n=1.8$ for different values of $U_0$. In the inset the results are shown for two different values of $\mathrm{\Delta }n=1.8$ and $\mathrm{\Delta }n=0$ for $\sigma =100$.

Figure 8

Snapshots of the density for the KH instability at times (a) $t=0.0$ (initial condition), (b) $t=3.31$, (c) $t=6.62$, (d) $t=15.47$, for the case with $\sigma =100, \mathrm{\Delta }n=0, U_0=0.6c$. The white lines show the magnetic field lines.

Figure 9

Reconnection rate vs time for the case $\mathrm{\Delta }n=1.8$ and $U_0=0.2c$ for different values of $\sigma$.

Figure 10

$R\left(t\right)$ at time $t=23.2$ for different values of $\sigma$ based on the numerical results (red symbols). Best fitting curves using the proportionality relation for the Sweet-Parker (dashed-dotted blue) and Petschek (dashed red) models are shown.

Figure 11

Snapshots of the 3D magnetic reconnection in a stellar flare due to the shear flow at times (a) $t=0$ (initial condition), (b) $t=19.86$, and (c) $t=43.02$. The colors of the magnetic lines show the magnitude of the magnetic field, where blue to red show low to high values. On the outer boundary surfaces of the domain the colors indicate the value of the vorticity which is indicated in the color bar.

Figure 12

Projection of the results presented in Fig. 11 onto the $xz$ plane. Times and colors are the same as Fig. 11.